edu-2013-05-mlc-exam-sol.pdf

《edu-2013-05-mlc-exam-sol.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、May2013MLCSolutions1.Key:DLetkbethepolicyyear,sothatthemortalityrateduringthatyearisq.Theobjectiveisto30+−k1determinethesmallestvalueofksuchthatkk−1vpPvp()kk−−130(100030)<()1303q01+k−(1000)Pv0.0068529+k29+>kk51>22Therefore,the

2、smallestvaluethatmeetstheconditionis23.2.Key:AExpectedPresentValueofBenefits:235,000(10/1,000)/1.067,500(15/1,000)/1.06++10,000(18/1,000)/1.06=++=47.17100.12151.13298.42.ExpectedPresentValueofPremiums:2[1(870/1,000)/1.06(701/1,000)/1.06]++PP=2.4446.Theannualpremi

3、umisP=298.42/2.4446=1223.Key:CTherearethreewaystoapproachthisproblem.Inallcases,letπdenotethebenefitpremium.,ThefirstapproachisanintuitiveresultbasedonthesolutiontoSampleQuestion309.Thekeyisthatinadditiontothepureendowment,thereisabenefitequalinvaluetoatemporarya

4、nnuityduewithannualpaymentπ.However,iftheinsuredsurvivesthe20years,thevalueoftheannuityisnotreceived.ππaE=+100,000a−paπ.40:20204040:202040206%Basedonthisequation,20100,000E100,000v100,000100,0002040π=====2,565paas38.9932040202020Thesecondapproachusesr

5、andomvariablestoderivetheexpectedpresentvalueofthereturnofpremiumbenefit.LetKbethecurtatefuturelifetimeof(40).ThepresentvaluerandomvariableisthenK+1πsv,2K<0Y=K+10,K≥20πaK,2<0K+1=0,K≥20πaK−<0,20K+1=.ππaaK−≥,202020Thefirsttermistherandomvariablet

6、hatcorrespondstoa20-yeartemporaryannuity.Thesecondtermistherandomvariablethatcorrespondstoapaymentwithapresentvalueofπa20contingentonsurviving20years.Theexpectedpresentvalueisthenππap−a.40:20204020Thethirdapproachtakesthemoststeps.1919k+1kk++11(1+−i)1ππaE

7、40:20=+100,0002040vsk+1kkq40=+100,00020E40πvq40kk==00d19ππk+11=+100,00020Eq40kk40−vq40=+100,00020E40()20q40−A40:20ddk=0π20=+100,00020Eqd40()2040−1+a40:20+v20p40d201−v=100,000Ea+−ππp204040:202040d=+100,000Eaππ−pa.204040:202040206%4.Key:CThereare

8、fourcareerpathsJoecouldfollow.Eachisoftheform:ppp()()(transitionprobability1)(transitionprobability2)353637wherethesurvivalprobabilitiesdependoneachyear’semplo